MathDB

2

Part of 2008 China Team Selection Test

Problems(7)

Chinese TST 2008 P2

Source:

4/3/2008
The sequence {xn} \{x_{n}\} is defined by x_{1} \equal{} 2,x_{2} \equal{} 12, and x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}, (n \equal{} 1,2,\ldots). Let p p be an odd prime number, let q q be a prime divisor of xp x_{p}. Prove that if q≠2,3, q\neq2,3, then q\geq 2p \minus{} 1.
inductionnumber theoryalgebraSequencerecurrence relation
Integer

Source: Chinese TST

4/5/2008
Let n>1 n > 1 be an integer, and n n can divide 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)}, let p1,p2,⋯ ,pk p_{1},p_{2},\cdots,p_{k} be all distinct prime divisors of n n. Show that \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}} is an integer. ( where ϕ(n) \phi(n) is defined as the number of positive integers ≤n \leq n that are relatively prime to n n.)
modular arithmeticnumber theoryrelatively primenumber theory proposed
Game

Source: Chinese TST

4/11/2008
In a plane, there is an infinite triangular grid consists of equilateral triangles whose lengths of the sides are equal to 1 1, call the vertices of the triangles the lattice points, call two lattice points are adjacent if the distance between the two points is equal to 1; 1; A jump game is played by two frogs A,B, A,B, "A jump" is called if the frogs jump from the point which it is lying on to its adjacent point, " A round jump of A,B A,B" is called if first A A jumps and then B B by the following rules: Rule (1): A A jumps once arbitrarily, then B B jumps once in the same direction, or twice in the opposite direction; Rule (2): when A,B A,B sits on adjacent lattice points, they carry out Rule (1) finishing a round jump, or A A jumps twice continually, keep adjacent with B B every time, and B B rests on previous position; If the original positions of A,B A,B are adjacent lattice points, determine whether for A A and B B,such that the one can exactly land on the original position of the other after a finite round jumps.
calculusintegrationrotationcombinatorics proposedcombinatorics
Set exists where subsets satisfying a condition are small

Source: Chinese TST

4/5/2008
Prove that for arbitary integer n>16 n > 16, there exists the set S S that contains n n positive integers and has the following property:if the subset A A of S S satisfies for arbitary a,a'\in A, a\neq a', a \plus{} a'\notin S holds, then ∣A∣≤4n. |A|\leq4\sqrt n.
combinatorics proposedcombinatorics
Easy inequality

Source: Chinese TST

4/6/2008
Let x,y,z x,y,z be positive real numbers, show that \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.
inequalitiesalgebrapolynomialinequalities proposed
If and only if

Source: Chinese TST

4/6/2008
For a given integer n≥2, n\geq 2, determine the necessary and sufficient conditions that real numbers a1,a2,⋯ ,an, a_{1},a_{2},\cdots, a_{n}, not all zero satisfy such that there exist integers 0<x1<x2<⋯<xn, 0<x_{1}<x_{2}<\cdots<x_{n}, satisfying a_{1}x_{1}\plus{}a_{2}x_{2}\plus{}\cdots\plus{}a_{n}x_{n}\geq 0.
inequalities proposedinequalities
Irreducible polynomial

Source: Chinese TST

4/9/2008
Prove that for all n≥2, n\geq 2, there exists n n-degree polynomial f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n} such that (1) a1,a2,⋯ ,an a_{1},a_{2},\cdots, a_{n} all are unequal to 0 0; (2) f(x) f(x) can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers x,∣f(x)∣ x, |f(x)| isn't prime numbers.
algebrapolynomialmodular arithmeticabsolute valuealgebra proposed